Section 7–7 / Relative Stability Analysis 471
It is noted, however, that in practical design problems the phase margin and gain
margin are more frequently specified than the resonant peak magnitude to indicate the
degree of damping in a system.
Correlation between Step Transient Response and Frequency Response in
the Standard Second-Order System. The maximum overshoot in the unit-step re-
sponse of the standard second-order system, as shown in Figure 7–73, can be exactly
correlated with the resonant peak magnitude in the frequency response. Hence, essen-
tially the same information about the system dynamics is contained in the frequency re-
sponse as is in the transient response.
For a unit-step input, the output of the system shown in Figure 7–73 is given by Equa-
tion (5–12), or
where
(7–19)
On the other hand, the maximum overshoot Mpfor the unit-step response is given by
Equation (5–21), or
(7–20)
This maximum overshoot occurs in the transient response that has the damped natural
frequency The maximum overshoot becomes excessive for values of
z<0.4.
Since the second-order system shown in Figure 7–73 has the open-loop transfer function
for sinusoidal operation, the magnitude of G(jv)becomes unity when
which can be obtained by equating @G(jv)@to unity and solving for v. At this frequency,
the phase angle of G(jv)is
Thus, the phase margin gis
(7–21)
Equation (7–21) gives the relationship between the damping ratio zand the phase margin
g. (Notice that the phase margin gis a function onlyof the damping ratio z.)
=tan-^1
2 z
321 + 4 z^4 - 2 z^2
= 90 °- tan-^1
321 + 4 z^4 - 2 z^2
2 z
g= 180 °+ /G(jv)
/G(jv)=-/jv-/jv+ 2 zvn=- 90 °-tan-^1
321 + 4 z^4 - 2 z^2
2 z
v=vn 321 + 4 z^4 - 2 z^2
G(s)=
v^2 n
sAs+ 2 zvnB
vd=vn 21 - z^2.
Mp=e-Az^21 - z
(^2) Bp
vd=vn 21 - z^2
c(t)= 1 - e-zvn^ tacosvd t+
z
21 - z^2
sinvd tb, fort 0